Entire solutions for several general quadratic trinomial differential difference equations (Q2669041)

Let \(\alpha,c\in\mathbb{C}\setminus\{0\}\) such that \(\alpha^{2} \neq 1\), and let \(g(z)\) be a polynomial. In the paper it is shown that if the difference equation \[ f(z+c)^{2}+2 \alpha f(z) f(z+c)+f(z)^{2}=e^{g(z)} \] admits a transcendental entire solution \(f(z)\) of finite order, then \(g(z)\) must be of the form \(g(z)=a z+b\), where \(a, b \in \mathbb{C}\). Moreover either \[ f(z)=\frac{1}{\sqrt{2}}\left(A_{1} \eta+A_{2} \eta^{-1}\right) e^{\frac{1}{2}(a z+b)}, \] where \(\eta \in \mathbb{C}\setminus\{0\}\) and \(a, c, A_{1}, A_{2}, \eta\) satisfy \[ e^{\frac{1}{2} a c}=\frac{A_{2} \eta+A_{1} \eta^{-1}}{A_{1} \eta+A_{2} \eta^{-1}}, \] or \[ f(z)=\frac{1}{\sqrt{2}}\left(A_{1} e^{a_1 z+b_{1}}+A_{2} e^{a_{2} z+b_{2}}\right) \] where \(a_{j}, b_{j} \in \mathbb{C},(j=1,2)\) satisfy \[ a_{1} \neq a_{2}, \quad g(z)=\left(a_{1}+a_{2}\right) z+b_{1}+b_{2}=a z+b \] and \[ e^{a_{1} c}=\frac{A_{2}}{A_{1}}, \quad e^{a_2 c}=\frac{A_{1}}{A_{2}}, \quad e^{a c}=1. \] In addition, the authors obtain a similar result for transcendental finite-order entire solutions of the equation \[ f(z+c)^{2}+2 \alpha f'(z) f(z+c)+f'(z)^{2}=e^{g(z)}. \]